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Simplifying 8z + 22 = 3(3z + 11) * z Reorder the terms: 22 + 8z = 3(3z + 11) * z Reorder the terms: 22 + 8z = 3(11 + 3z) * z Reorder the terms for easier multiplication: 22 + 8z = 3z(11 + 3z) 22 + 8z = (11 * 3z + 3z * 3z) 22 + 8z = (33z + 9z2) Solving 22 + 8z = 33z + 9z2 Solving for variable 'z'. Combine like terms: 8z + -33z = -25z 22 + -25z + -9z2 = 33z + 9z2 + -33z + -9z2 Reorder the terms: 22 + -25z + -9z2 = 33z + -33z + 9z2 + -9z2 Combine like terms: 33z + -33z = 0 22 + -25z + -9z2 = 0 + 9z2 + -9z2 22 + -25z + -9z2 = 9z2 + -9z2 Combine like terms: 9z2 + -9z2 = 0 22 + -25z + -9z2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -2.444444444 + 2.777777778z + z2 = 0 Move the constant term to the right: Add '2.444444444' to each side of the equation. -2.444444444 + 2.777777778z + 2.444444444 + z2 = 0 + 2.444444444 Reorder the terms: -2.444444444 + 2.444444444 + 2.777777778z + z2 = 0 + 2.444444444 Combine like terms: -2.444444444 + 2.444444444 = 0.000000000 0.000000000 + 2.777777778z + z2 = 0 + 2.444444444 2.777777778z + z2 = 0 + 2.444444444 Combine like terms: 0 + 2.444444444 = 2.444444444 2.777777778z + z2 = 2.444444444 The z term is 2.777777778z. Take half its coefficient (1.388888889). Square it (1.929012346) and add it to both sides. Add '1.929012346' to each side of the equation. 2.777777778z + 1.929012346 + z2 = 2.444444444 + 1.929012346 Reorder the terms: 1.929012346 + 2.777777778z + z2 = 2.444444444 + 1.929012346 Combine like terms: 2.444444444 + 1.929012346 = 4.37345679 1.929012346 + 2.777777778z + z2 = 4.37345679 Factor a perfect square on the left side: (z + 1.388888889)(z + 1.388888889) = 4.37345679 Calculate the square root of the right side: 2.091281136 Break this problem into two subproblems by setting (z + 1.388888889) equal to 2.091281136 and -2.091281136.Subproblem 1
z + 1.388888889 = 2.091281136 Simplifying z + 1.388888889 = 2.091281136 Reorder the terms: 1.388888889 + z = 2.091281136 Solving 1.388888889 + z = 2.091281136 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.388888889' to each side of the equation. 1.388888889 + -1.388888889 + z = 2.091281136 + -1.388888889 Combine like terms: 1.388888889 + -1.388888889 = 0.000000000 0.000000000 + z = 2.091281136 + -1.388888889 z = 2.091281136 + -1.388888889 Combine like terms: 2.091281136 + -1.388888889 = 0.702392247 z = 0.702392247 Simplifying z = 0.702392247Subproblem 2
z + 1.388888889 = -2.091281136 Simplifying z + 1.388888889 = -2.091281136 Reorder the terms: 1.388888889 + z = -2.091281136 Solving 1.388888889 + z = -2.091281136 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.388888889' to each side of the equation. 1.388888889 + -1.388888889 + z = -2.091281136 + -1.388888889 Combine like terms: 1.388888889 + -1.388888889 = 0.000000000 0.000000000 + z = -2.091281136 + -1.388888889 z = -2.091281136 + -1.388888889 Combine like terms: -2.091281136 + -1.388888889 = -3.480170025 z = -3.480170025 Simplifying z = -3.480170025Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.702392247, -3.480170025}
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